Integer Representations in IEEE 754, Posit, and Takum Arithmetics

TL;DR

This paper rigorously analyzes the integral representation capabilities of various floating-point formats, including IEEE 754, bfloat16, posit, takum, and OFP8, revealing that takums offer the most effective and compatible solutions.

Contribution

It provides the first formal derivations and proofs of integer representation properties across multiple floating-point formats, comparing their effectiveness.

Findings

OFP8 has mixed performance in integer representation.

bfloat16 generally underperforms compared to other formats.

Takums consistently match or outperform others and maintain IEEE 754 compatibility.

Abstract

Although not primarily designed for this purpose, floating-point numbers are often used to represent integral values, with some applications explicitly relying on this capability. However, the integral representation properties of IEEE 754 floating-point numbers have not yet been formally investigated. Recently, the bfloat16, posit and takum machine number formats have been proposed as alternatives to IEEE 754, while OCP 8-bit floating point (OFP8) types (E4M3 and E5M2) have been introduced as 8-bit extensions of IEEE 754, albeit with slight deviations from the standard. It is therefore timely to evaluate IEEE 754 and to assess how effectively the new formats fulfil this function in comparison with the standard they aim to replace. This paper presents the first rigorous derivations and proofs of the integral representation capabilities of IEEE 754 floating-point numbers, OFP8, bfloat16, posits, and takums. We examine both the exact number of bits required to represent a given integer and the largest consecutive integer representable with a specified bit width. The results show that OFP8 yields mixed outcomes, bfloat16 generally underperforms, and posits fail to scale effectively, whereas takums consistently match or outperform the other formats, maintaining backward compatibility with IEEE 754.